Curriculum
Title Real Analysis (Code 09010501)
S.N 
Unit 
Content of the topics 
Learning objects 
Teaching Guidelines 
Methodology 
Time (Hrs) 
1 
one 
Riemann integral, Integrabililty of continuous and monotonic functions, The Fundamental theorem of integral calculus. Mean value theorems of integral calculus. Improper integrals and their convergence, Comparison tests, Abel’s and Dirichlet’s tests, Frullani’s integral, Integral as a function of a parameter. Continuity, Differentiability and integrability of an integral of a function of a parameter. Riemann integral, Integrabililty of continuous and monotonic functions, The Fundamental theorem of integral calculus. Mean value theorems of integral calculus. 
Students will be able to
1. Divisibility, G.C.D., L.C.M.. 
Lecture should be effective so that student can grasp the topics easily. 

15 
2 
Two 
Definition and examples of metric spaces metrics Definition and examples of metric spaces metrics Definition and examples of metric spaces metrics Definition and examples of metric spaces 
Students will be able to 1. Complete residue system , Quadratic residue, Euler’s ø function 2. Generalization of Fermat’s Theorem.Chinese remainder Theorem Legendre symbols, Gauss integer function,Chinese remainder theorem. 
Lecture should be effective so that student can grasp the topics easily. 

15 
3 
Three 
Continuous functions, uniform continuity, compactness for metric spaces, sequential compactness, BolzanoWeierstrass property, total boundedness, finite intersection property, continuity in relation with compactness, connectedness , components, continuity in relation with connectedness. Continuous functions, uniform continuity, compactness for metric spaces, sequential compactness, BolzanoWeierstrass property, total boundedness, finite intersection 
To make students familiar with Direct circular and hyperbolic functions and their properties, De Moivre’s Theorem and its applications. Inverse circular and hyperbolic functions and their properties, Logarithm of a complex quantity. Gregory series, Summation of Trigonometric series 
Lecture should be effective sothat student can grasp the topics easily. 

15 